As degrees of integration of semiconductor devices are increased and the sizes of semiconductor devices are decreased, serious problems with regard to the generation of leakage current are increasing. Thus, leakage estimation and reduction techniques have been one of the most important design factors in manufacturing an integrated circuit. The increasing amounts of leakage current not only prevent normal operation of the integrated circuit but also consume excessive driving power, and thus it is increasingly critical for the device performance of the integrated circuit. Particularly, battery-powered devices including integrated circuits, such as mobile and handheld electronics, have recently become widespread, and thus the excessive power consumption has been a critical factor for the performance of the battery-powered devices. For those reasons, the leakage estimation and reduction techniques have become much more important factors in designing the ICs.
Most of the leakage estimation and reduction techniques have focused on sub-threshold leakage due to lowering of power supply voltages and accompanying reductions of threshold voltages. However, with the recent high degrees of integration and the decreases in the sizes of ICs, the gate leakage current occurring at a gate electrode of the ICs, as well as the sub-threshold leakage current, has also become important factors in designing the ICs. Accordingly, accurate full chip leakage estimation is frequently required for a chip design of the ICs so as to estimate both of the sub-threshold leakage and the gate leakage. Particularly, in recent very large-scale integration (VLSI) chips, tunneling of carriers through a gate insulation layer may frequently occur due to a reduced thickness of the gate insulation layer, and thus the estimation of the gate leakage necessarily needs to be considered in designing the VLSI chips.
Various estimation models for estimating the full chip leakage have been suggested for the last few years. It is well known that the full chip leakage in a chip may be influenced by various factors, such as process parameters, for example, a line width and a critical dimension (CD) of a pattern, and environmental factors, for example, channel temperature, power supply voltage (Vdd), circuit topology and an allowable load. Thus, a specific estimation model for estimating the full chip leakage with respect to a specific factor does not give sufficiently accurate information on the full chip leakage in a chip.
Therefore, statistical models have been suggested for the full chip leakage estimation model in which conventional experimental results are operated by well-known statistical methods and the full chip leakage is estimated in consideration of all of the above factors including the process parameters and the environmental factors. Particularly, a log normal estimation model has been most widely used for the full chip leakage estimation model among the above statistical models. According to the lognormal estimation model, every factor of the process parameters and the environmental factors functions as a random variable for a probability density function (PDF), and thus a lognormal PDF is generated with respect to each of the various process parameters and the environmental factors. Then, each of the lognormal PDFs is functionally summarized into a single composite probability function, and an optimal point is determined by using the single composite probability function where the full chip leakage is minimized in consideration of all of the above factors. The values of the random variables at the optimal point of the single composite probability function are regarded as optimal design factors for minimizing the full chip leakage in a chip. The random variable of the lognormal PDF is an exponential of the random variable of a normal PDF, and thus the multiplication of the lognormal random variables is also distributed in accordance with a lognormal distribution. For those reasons, the lognormal variable has been widely used for statistical estimation and analysis for a situation where the statistical error is strongly influenced by multiplication of environmental factors.
However, the statistical estimation model for the full chip leakage using the lognormal PDF (hereinafter referred to as lognormal leakage estimation model) has a critical demerit of computational complexity. Particularly, the computational complexity of the lognormal leakage estimation model may be geometrically increased as the number of the environmental factors related to the statistical error of the lognormal leakage estimation model. Thus, the lognormal leakage estimation model has many limitations when being applied to a circuit design.
According to a conventional lognormal leakage estimation model, a semiconductor chip is divided into a plurality of estimation regions by a grid and the lognormal PDF is generated at each of the estimation regions. Then, each of the lognormal PDFs at the estimation regions is statistically summed up in consideration of spatial correlation between the estimation regions, to thereby generate a full chip lognormal PDF indicating probability of a leakage current from a whole chip on the wafer.
Particularly, the probability of the leakage current at an arbitrary cell l of the chip is expressed as follows when the leakage current is influenced by an arbitrary environmental factor i.
                              I          i          l                =                  ⅇ                                    a              0                        +                                          ∑                                  j                  =                  1                                n                            ⁢                                                a                  j                                ⁢                                  P                  j                                                      +                                          a                                  n                  +                  1                                            ⁢              R                                                          (        1        )            
The polynomial term in the above exponential equation (1) indicates a normal distribution having an average of a0 and a variation of Σaj2 on condition that the parameters Pj and R may be distributed as a standard normal distribution N(0,1) Accordingly, the leakage current at an arbitrary cell is calculated by equation (1) on a whole chip, and thus is distributed on the whole chip as a lognormal distribution.
In the above equation (1), the parameter Pj is a global parameter indicating outer environment factors at each of the chips. That is, the parameter Pj includes a random variable indicating random variation in a manufacturing process for a semiconductor device. The random variable of the individual chip includes a variation of a die-to-die parameter in which the parameter may be uniformly varied on the whole chip and a variation of a within-die-parameter in which the parameter may be non-uniformly varied on the whole chip, and thus the variation of the within-die-parameter may be expressed by spatial correlation.
In addition, the parameter R in the above equation (1) is a local parameter as a single random variable into which various independent variables are grouped. The independent variable has an individual effect on the current leakage at a local area of the chip irrespectively of other variables. A plurality of the independent variables can be treated as a single variable without any computational error in equation (1), and thus the number of the variables is remarkably reduced in conducting equation (1) to thereby significantly reduce the computational load.
Further, a0, aj, an+1, in equation (1) are fitting coefficients for transforming the distribution of the probability function expressed by equation (1) into a normal distribution and indicate correlation between the global parameter or the local parameter and the leakage current. When the probability function of equation (1) including the global parameter and the local parameter is transformed into the normal distribution, multiplication of the lognormal distribution is transformed into summation of the standard normal distribution, to thereby significantly reduce the computational load when performing a leakage estimation process.
Since the leakage current at an arbitrary cell is calculated by equation (1) with respect to a specific input factor, an average leakage current over a whole chip (full chip leakage current) with respect to all of the input factors may be calculated by the following equation (2).
                              I          avg                =                                            ∑                              l                =                1                            p                        ⁢                          (                                                ∑                                      i                    =                    1                                    m                                ⁢                                                      Pr                    i                                    ⁢                                      I                    i                    l                                                              )                                =                                    ∑                              l                =                1                            p                        ⁢                          (                                                ∑                                      i                    =                    1                                    m                                ⁢                                                      Pr                    i                                    ⁢                                      ⅇ                                                                  a                        0                                            +                                                                        ∑                                                      j                            =                            1                                                    n                                                ⁢                                                                              a                            j                                                    ⁢                                                      p                            j                                                                                              +                                                                        a                                                      n                            +                            1                                                                          ⁢                        R                                                                                                        )                                                          (        2        )            
In the above equation (2), Pri indicates a probability that an arbitrary input factor i may be applied to an arbitrary cell l of the chip and the number of the input factors is m and number of the cells on the chip is p.
As shown in the above equation (2), the full chip leakage current is calculated through the summation of the lognormal distribution and Wilkinson's method has been widely used for the summation of the lognormal distribution.
According to Wilkinson's method, a single lognormal distribution, which may be statistically equivalent to the summation of a number of the individual lognormal distributions, is generated by using a first moment and a second moment as follows.eY1+eY2+ . . . +eYn=eZ  (3)
In the above equation (3), each of the probability distributions Yi includes a normal distribution having an average of μYi and a standard variation of σYi. Supposing that the probability distribution Z of the equivalent lognormal probability function is a normal distribution having an average of μz and a standard variation of σz, the first and second moments of the equivalent lognormal distribution are expressed as follows.
                              μ          1                =                              E            ⁡                          [                                                ⅇ                                      Y                    1                                                  +                                  ⅇ                                      Y                    2                                                  +                …                +                                  ⅇ                                      Y                    n                                                              ]                                =                      ⅇ                                          μ                z                            +                                                σ                  2                  2                                2                                                                        (        4        )                                                                                    μ                2                            =                            ⁢                              E                ⁡                                  [                                                            (                                                                        ⅇ                                                      Y                            1                                                                          +                                                  ⅇ                                                      Y                            2                                                                          +                                                  …                          ⁢                                                                                                          ⁢                                                      ⅇ                                                          Y                              n                                                                                                                          )                                        2                                    ]                                                                                                        =                            ⁢                              ⅇ                                                      2                    ⁢                                          μ                      z                                                        +                                      2                    ⁢                                          σ                      z                      2                                                                                                                                              =                            ⁢                                                                    ∑                                          i                      =                      1                                        n                                    ⁢                                      ⅇ                                                                  2                        ⁢                                                  μ                          Yi                                                                    +                                              2                        ⁢                                                  σ                          Yi                          2                                                                                                                    +                                  2                  ⁢                                                            ∑                                              i                        =                        1                                                                    n                        -                        1                                                              ⁢                                          (                                                                        ∑                                                      j                            =                                                          i                              +                              1                                                                                n                                                ⁢                                                  ⅇ                                                                                    μ                              Yi                                                        +                                                          μ                              Yj                                                        +                                                                                                                                                                                                                                              σ                                        Yi                                        2                                                                            +                                                                              σ                                        Yj                                        2                                                                            +                                                                                                                                                                                                                                                  2                                      ⁢                                                                              r                                        ij                                                                            ⁢                                                                              σ                                        Yi                                                                            ⁢                                                                              σ                                        Yj                                                                                                                                                                                                        2                                                                                                                          )                                                                                                                              (        5        )            
In the above equation (5), rij indicates a correlation coefficient between the different probability distributions Yi and Yj.
Solutions of the simultaneous equations (4) and (5) provide the average and the standard deviation of the probability distribution Z, to thereby determine the lognormal distribution Z. Then, the full chip leakage current is estimated by the lognormal distribution Z.
However, as shown in the second term of equation (5), the correlation coefficients between the different probability distributions usually cause a tremendous computational load in operating the statistical summation of equation (5), to thereby significantly increase the operation complexity of equation (5).
The operation complexity of the lognormal distribution model is determined by the number of the individual lognormal distributions and is expressed as O(N2). N is the number of the individual lognormal distributions that are to be statistically summed up in calculating the first and second moments. According to the conventional lognormal distribution model for estimating the full chip current leakage, a number of (NC*M) of the independent lognormal distributions are generated when the full chip includes a number of NC of the cells and the kinds of the current leakage is a number of M and the operation complexity of Wilkinson's method is calculated as O((NC*M)2). Therefore, the operation complexity is extremely increased when Wilkinson's method is performed on a relatively large size of an electric circuit and finally exceeds the operation capability of the current computer systems. Particularly, when the degree of integration of semiconductor devices is increased, and thus a plurality of electric circuits is integrated into a small area of a substrate, the operation complexity of Wilkinson's method is further increased and finally Wilkinson's method cannot be substantially applied for estimating the full chip leakage.
For the above reasons, there is still a need for an improved estimation model for estimating the full chip leakage current in which the full chip leakage current is accurately estimated with a relatively low operation complexity.